Theorem isdivrngo 37990

Description: The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
isdivrngo	⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Proof of Theorem isdivrngo

Dummy variables 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5087	. . . . 5 ⊢ (𝐺DivRingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps)
2		df-drngo 37989	. . . . . . 7 ⊢ DivRingOps = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ RingOps ∧ (𝑦 ↾ ((ran 𝑥 ∖ {(GId‘𝑥)}) × (ran 𝑥 ∖ {(GId‘𝑥)}))) ∈ GrpOp)}
3	2	relopabiv 5755	. . . . . 6 ⊢ Rel DivRingOps
4	3	brrelex1i 5667	. . . . 5 ⊢ (𝐺DivRingOps𝐻 → 𝐺 ∈ V)
5	1, 4	sylbir 235	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps → 𝐺 ∈ V)
6	5	anim1i 615	. . 3 ⊢ ((⟨𝐺, 𝐻⟩ ∈ DivRingOps ∧ 𝐻 ∈ 𝐴) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
7	6	ancoms 458	. 2 ⊢ ((𝐻 ∈ 𝐴 ∧ ⟨𝐺, 𝐻⟩ ∈ DivRingOps) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
8		rngoablo2 37949	. . . . 5 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)
9		elex 3457	. . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V)
10	8, 9	syl 17	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V)
11	10	ad2antrl 728	. . 3 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐺 ∈ V)
12		simpl 482	. . 3 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐻 ∈ 𝐴)
13	11, 12	jca 511	. 2 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
14		df-drngo 37989	. . . 4 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
15	14	eleq2i 2823	. . 3 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)})
16		opeq1 4820	. . . . . 6 ⊢ (𝑔 = 𝐺 → ⟨𝑔, ℎ⟩ = ⟨𝐺, ℎ⟩)
17	16	eleq1d 2816	. . . . 5 ⊢ (𝑔 = 𝐺 → (⟨𝑔, ℎ⟩ ∈ RingOps ↔ ⟨𝐺, ℎ⟩ ∈ RingOps))
18		rneq 5871	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
19		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
20	19	sneqd 4583	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → {(GId‘𝑔)} = {(GId‘𝐺)})
21	18, 20	difeq12d 4072	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (ran 𝑔 ∖ {(GId‘𝑔)}) = (ran 𝐺 ∖ {(GId‘𝐺)}))
22	21	sqxpeqd 5643	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
23	22	reseq2d 5923	. . . . . 6 ⊢ (𝑔 = 𝐺 → (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
24	23	eleq1d 2816	. . . . 5 ⊢ (𝑔 = 𝐺 → ((ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp ↔ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
25	17, 24	anbi12d 632	. . . 4 ⊢ (𝑔 = 𝐺 → ((⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp) ↔ (⟨𝐺, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
26		opeq2 4821	. . . . . 6 ⊢ (ℎ = 𝐻 → ⟨𝐺, ℎ⟩ = ⟨𝐺, 𝐻⟩)
27	26	eleq1d 2816	. . . . 5 ⊢ (ℎ = 𝐻 → (⟨𝐺, ℎ⟩ ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps))
28		reseq1 5917	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
29	28	eleq1d 2816	. . . . 5 ⊢ (ℎ = 𝐻 → ((ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp ↔ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
30	27, 29	anbi12d 632	. . . 4 ⊢ (ℎ = 𝐻 → ((⟨𝐺, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
31	25, 30	opelopabg 5473	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)} ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
32	15, 31	bitrid 283	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
33	7, 13, 32	pm5.21nd 801	1 ⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3894  {csn 4571  ⟨cop 4577   class class class wbr 5086  {copab 5148   × cxp 5609  ran crn 5612   ↾ cres 5613  ‘cfv 6476  GrpOpcgr 30461  GIdcgi 30462  AbelOpcablo 30516  RingOpscrngo 37934  DivRingOpscdrng 37988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-1st 7916  df-2nd 7917  df-rngo 37935  df-drngo 37989

This theorem is referenced by:  zrdivrng  37993  isdrngo1  37996